| L(s) = 1 | + (−0.437 − 1.34i)2-s + (3.50 − 3.15i)3-s + (−1.61 + 1.17i)4-s + (−3.00 − 5.21i)5-s + (−5.78 − 3.33i)6-s + (1.44 + 13.7i)7-s + (2.28 + 1.66i)8-s + (1.39 − 13.2i)9-s + (−5.69 + 6.32i)10-s + (4.15 − 9.32i)11-s + (−1.96 + 9.23i)12-s + (−0.618 − 2.90i)13-s + (17.8 − 7.94i)14-s + (−27.0 − 8.78i)15-s + (1.23 − 3.80i)16-s + (8.78 + 19.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s + (1.16 − 1.05i)3-s + (−0.404 + 0.293i)4-s + (−0.601 − 1.04i)5-s + (−0.963 − 0.556i)6-s + (0.206 + 1.96i)7-s + (0.286 + 0.207i)8-s + (0.154 − 1.46i)9-s + (−0.569 + 0.632i)10-s + (0.377 − 0.847i)11-s + (−0.163 + 0.769i)12-s + (−0.0475 − 0.223i)13-s + (1.27 − 0.567i)14-s + (−1.80 − 0.585i)15-s + (0.0772 − 0.237i)16-s + (0.516 + 1.16i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.127 + 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.127 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.894168 - 1.01617i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.894168 - 1.01617i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.437 + 1.34i)T \) |
| 31 | \( 1 + (30.8 - 3.15i)T \) |
| good | 3 | \( 1 + (-3.50 + 3.15i)T + (0.940 - 8.95i)T^{2} \) |
| 5 | \( 1 + (3.00 + 5.21i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (-1.44 - 13.7i)T + (-47.9 + 10.1i)T^{2} \) |
| 11 | \( 1 + (-4.15 + 9.32i)T + (-80.9 - 89.9i)T^{2} \) |
| 13 | \( 1 + (0.618 + 2.90i)T + (-154. + 68.7i)T^{2} \) |
| 17 | \( 1 + (-8.78 - 19.7i)T + (-193. + 214. i)T^{2} \) |
| 19 | \( 1 + (-14.3 - 3.05i)T + (329. + 146. i)T^{2} \) |
| 23 | \( 1 + (3.25 - 4.48i)T + (-163. - 503. i)T^{2} \) |
| 29 | \( 1 + (37.8 - 12.3i)T + (680. - 494. i)T^{2} \) |
| 37 | \( 1 + (-13.9 - 8.02i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (5.51 - 6.11i)T + (-175. - 1.67e3i)T^{2} \) |
| 43 | \( 1 + (-2.56 + 12.0i)T + (-1.68e3 - 752. i)T^{2} \) |
| 47 | \( 1 + (-7.52 + 23.1i)T + (-1.78e3 - 1.29e3i)T^{2} \) |
| 53 | \( 1 + (-2.87 - 0.302i)T + (2.74e3 + 584. i)T^{2} \) |
| 59 | \( 1 + (23.0 + 25.5i)T + (-363. + 3.46e3i)T^{2} \) |
| 61 | \( 1 - 4.33iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (47.8 + 82.8i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-1.02 + 9.78i)T + (-4.93e3 - 1.04e3i)T^{2} \) |
| 73 | \( 1 + (26.5 - 59.6i)T + (-3.56e3 - 3.96e3i)T^{2} \) |
| 79 | \( 1 + (-50.0 - 112. i)T + (-4.17e3 + 4.63e3i)T^{2} \) |
| 83 | \( 1 + (98.2 + 88.4i)T + (720. + 6.85e3i)T^{2} \) |
| 89 | \( 1 + (15.2 + 20.9i)T + (-2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (10.7 - 7.82i)T + (2.90e3 - 8.94e3i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.33009462070338349442149109341, −12.93227512367262820816002289093, −12.42437051429498594142172973090, −11.53640268407959966531646417614, −9.222838000230625008279118698290, −8.611330698395864994476632552298, −7.897334388389000642890211304375, −5.62784514360131040484089630470, −3.32953063701047154489512453982, −1.70372767614843295425940910524,
3.50388807831042511683308455253, 4.47414343028231482102332253237, 7.25138856224828802362065689848, 7.56906836396555898906027245566, 9.386485701084800827409491727843, 10.16688281351398923963311870777, 11.21423336916249515773016531637, 13.55482502418897620260700356298, 14.39304069915916937977323239229, 14.83484020998800552687654584561
